# How to summarize measurement variability for repeated measures?

I have a data set where different measures are taken repeatedly (3) times for multiple samples, like so:

   A B C
X  ...
Y  ...


So, for people X, Y, etc I have measures A, B, C and so on. Each of A, B, C are measured 3 times for each sample. That is, for each person, I have 3 measures of A (A1, A2, A3), 3 measures of B (B1, B2, B3), for each of the columns.

What I'd like to know is, how can I summarize the variability or the reliability of each column? So, over the entire data set, how reliable is the measurement for A?

Thanks for the help!

## 1 Answer

the sample variance of the measurements A1, A2, A3, for a person X is:

$$\sigma_X(A) = \frac{1}{2} \sum_{i=1}^{3} (A_{i,X} - \bar{A}_X)^2$$

where $\bar{A}_X$ is the arithmetic mean of the measurement A for person X. Now you can look at the $\sigma_X(A)$ values across all samples (people) in your study, and for example consider mean value $\bar{\sigma}(A)$ to quantify the reliability of measurements A.

$$\bar{\sigma}(A) = \frac{1}{N} \sum_X \sigma_X(A)$$

Another way of looking at it is if you consider each $A_i$, for $i=1,2,3$, across all samples. Define $\bar{A}_i$ as the mean value of the $A_i$ measurements across samples. This allows us to characterize the standard deviation $\sigma_i(A)$ of a particular measurement $A_i$ across samples.

$$\sigma_i(A) = \frac{1}{2} \sum_{X} (A_{i,X} - \bar{A}_i)^2$$

The average $\sigma_i(A)$ for all $i=1,2,3$ indicates the average variability of the measurement A.

Hope it was helpful!

• Thanks, that's great. I'm also comparing another set of measurements, so given the average variance I can then ask questions like "How confident am I that the two means are different?" Feb 12, 2014 at 17:09